On the toroidal background \(T^{2n}\) , the geometrical space group is the \(SL(2n,\mathbb {Z})\) 
modular group.
For example, the CP transformation on \(T^2\)  is identified with the outer automorphism
of \(SL(2, \mathbb {Z})\)  [1]}, [2]}.
It is interesting to ask whether CP is embedded into the symplectic modular transformation on CY threefolds.
The CP transformation we discussed corresponds to the
following transformation for the period integrals,
\(\Pi =\begin{pmatrix}X^0\\X^i\\{\cal F}_0\\{\cal F}_i\\\end{pmatrix}\rightarrow \pm \begin{pmatrix}X^0\\-\bar{X}^i\\-\bar{{\cal F}}_0\\\bar{{\cal F}}_i\\\end{pmatrix}= \pm \begin{pmatrix}1 & 0 & 0 & 0\\0 & -\bf {1} & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & \bf {1}\\\end{pmatrix}\bar{\Pi }.\) 
